direct product, p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C2×C4⋊1D4, C42⋊18C22, C23.8C23, C22.24C24, C24.15C22, C4⋊1(C2×D4), (C2×C4)⋊7D4, (C2×C42)⋊11C2, (C22×D4)⋊5C2, (C2×D4)⋊12C22, C22.62(C2×D4), C2.10(C22×D4), (C2×C4).130C23, (C22×C4).123C22, SmallGroup(64,211)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C4⋊1D4
G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 441 in 249 conjugacy classes, 105 normal (5 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C23, C42, C22×C4, C2×D4, C2×D4, C24, C2×C42, C4⋊1D4, C22×D4, C2×C4⋊1D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C4⋊1D4, C22×D4, C2×C4⋊1D4
Character table of C2×C4⋊1D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ13 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ14 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ15 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ16 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | orthogonal lifted from D4 |
| ρ21 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | -2 | 2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ24 | 2 | 2 | -2 | 2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ25 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | orthogonal lifted from D4 |
| ρ26 | 2 | 2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ27 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ28 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | orthogonal lifted from D4 |
►On 32 points
(1 12)(2 9)(3 10)(4 11)(5 24)(6 21)(7 22)(8 23)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 18 31 6)(2 19 32 7)(3 20 29 8)(4 17 30 5)(9 14 25 22)(10 15 26 23)(11 16 27 24)(12 13 28 21)
(1 26)(2 25)(3 28)(4 27)(5 24)(6 23)(7 22)(8 21)(9 32)(10 31)(11 30)(12 29)(13 20)(14 19)(15 18)(16 17)
G:=sub<Sym(32)| (1,12)(2,9)(3,10)(4,11)(5,24)(6,21)(7,22)(8,23)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,18,31,6)(2,19,32,7)(3,20,29,8)(4,17,30,5)(9,14,25,22)(10,15,26,23)(11,16,27,24)(12,13,28,21), (1,26)(2,25)(3,28)(4,27)(5,24)(6,23)(7,22)(8,21)(9,32)(10,31)(11,30)(12,29)(13,20)(14,19)(15,18)(16,17)>;
G:=Group( (1,12)(2,9)(3,10)(4,11)(5,24)(6,21)(7,22)(8,23)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,18,31,6)(2,19,32,7)(3,20,29,8)(4,17,30,5)(9,14,25,22)(10,15,26,23)(11,16,27,24)(12,13,28,21), (1,26)(2,25)(3,28)(4,27)(5,24)(6,23)(7,22)(8,21)(9,32)(10,31)(11,30)(12,29)(13,20)(14,19)(15,18)(16,17) );
G=PermutationGroup([[(1,12),(2,9),(3,10),(4,11),(5,24),(6,21),(7,22),(8,23),(13,18),(14,19),(15,20),(16,17),(25,32),(26,29),(27,30),(28,31)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,18,31,6),(2,19,32,7),(3,20,29,8),(4,17,30,5),(9,14,25,22),(10,15,26,23),(11,16,27,24),(12,13,28,21)], [(1,26),(2,25),(3,28),(4,27),(5,24),(6,23),(7,22),(8,21),(9,32),(10,31),(11,30),(12,29),(13,20),(14,19),(15,18),(16,17)]])
C2×C4⋊1D4 is a maximal subgroup of
C42.413D4 C42.82D4 C24.24D4 C42.432D4 C42.112D4 M4(2)⋊12D4 C42.118D4 C42⋊9D4 (C2×C4)⋊2D8 (C2×C8)⋊20D4 C4⋊C4⋊7D4 C42⋊14D4 C24.219C23 C23.262C24 C23.328C24 C24.263C23 C23.333C24 C23.345C24 C42⋊20D4 C42⋊21D4 C42.171D4 C42⋊27D4 C42.194D4 C42⋊31D4 C23.569C24 C23.573C24 C24.411C23 C42⋊33D4 C42⋊47D4 C43⋊12C2 C43⋊15C2 C42.444D4 C42.240D4 M4(2)⋊7D4 C42.263D4 C42.266D4 C42.275D4 C2×D42 C22.87C25 C22.97C25 C22.132C25
C2×C4⋊1D4 is a maximal quotient of
C42⋊16D4 C23.333C24 C23.401C24 C23.439C24 C42⋊19D4 C42⋊20D4 C42.167D4 C23.556C24 C42⋊31D4 C42.196D4 C42⋊47D4 C43⋊12C2 C43⋊15C2 C42⋊19Q8 C42.360D4 C42.247D4 M4(2)⋊7D4 M4(2)⋊8D4 M4(2)⋊9D4 M4(2)⋊10D4 M4(2)⋊11D4 M4(2).20D4
Matrix representation of C2×C4⋊1D4 ►in GL5(ℤ)
| -1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | -1 | 2 | 0 | 0 |
| 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | -1 | 0 |
| -1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | -1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 |
| 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,-1,-1,0,0,0,2,1,0,0,0,0,0,0,-1,0,0,0,1,0],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0],[1,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1] >;
C2×C4⋊1D4 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes_1D_4
% in TeX
G:=Group("C2xC4:1D4"); // GroupNames label
G:=SmallGroup(64,211);
// by ID
G=gap.SmallGroup(64,211);
# by ID
G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650,158]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
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